what is k equal to in coulomb law

Proportionality constant in electrodynamics equations

Value of m	Units
8.987551 7923(fourteen)×x9	Northward·m2/Ctwo
fourteen.3996	eV·Å·due east −2
10−7	(N·stwo/C2)c 2

The Coulomb constant, the electric strength constant, or the electrostatic constant (denoted k _{due east} , thousand or M ) is a proportionality constant in electrostatics equations. In SI units it is equal to viii.987551 7923(fourteen)×10^nine kg⋅1000³⋅south⁻²⋅C^−two .^[1] Information technology was named after the French physicist Charles-Augustin de Coulomb (1736–1806) who introduced Coulomb's police.^{[2] [3]}

Value of the constant [edit]

The Coulomb constant is the constant of proportionality in Coulomb'south law,

${\displaystyle \mathbf {F} =k_{\text{eastward}}{\frac {Qq}{r^{two}}}\mathbf {\hat {eastward}} _{r}}$

where ê _r is a unit vector in the r -management.^[iv] In SI:

${\displaystyle k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}},}$

where ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity. This formula can be derived from Gauss' law,

${\displaystyle {\scriptstyle Due south}}$ ${\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}}$

Taking this integral for a sphere, radius r , centered on a betoken charge, the electric field points radially outwards and is normal to a differential surface element on the sphere with abiding magnitude for all points on the sphere.

${\displaystyle {\scriptstyle Due south}}$ ${\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} =|\mathbf {E} |\int _{S}dA=|\mathbf {E} |\times iv\pi r^{two}}$

Noting that E = F / q for some test accuse q ,

${\displaystyle {\brainstorm{aligned}\mathbf {F} &={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\chapeau {e}} _{r}=k_{\text{east}}{\frac {Qq}{r^{ii}}}\mathbf {\chapeau {e}} _{r}\\[8pt]\therefore k_{\text{due east}}&={\frac {one}{4\pi \varepsilon _{0}}}\end{aligned}}}$

Coulomb's law is an changed-foursquare law, and thereby similar to many other scientific laws ranging from gravitational pull to low-cal attenuation. This law states that a specified physical quantity is inversely proportional to the square of the altitude.

$${\displaystyle {\text{intensity}}={\frac {ane}{d^{ii}}}}$$

In some modernistic systems of units, the Coulomb constant k _east has an exact numeric value; in Gaussian units k _e = 1, in Lorentz–Heaviside units (also called rationalized) one thousand _eastward = 1 / 4π . This was previously true in SI when the vacuum permeability was defined as μ ₀ = 4π ×ten ^−seven H⋅g⁻¹ . Together with the speed of light in vacuum c , defined equally 299792 458 g/s, the vacuum permittivity ε ₀ tin can be written as

1 / μ ₀ c ² , which gave an exact value of^[five]

${\displaystyle {\begin{aligned}k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}={\frac {c^{2}\mu _{0}}{4\pi }}&=c^{ii}\times (x^{-vii}\ \mathrm {H{\cdot }m} ^{-one})\\&=8.987\ 551\ 787\ 368\ 1764\times ten^{9}~\mathrm {N{\cdot }m^{two}{\cdot }C^{-2}} .\terminate{aligned}}}$

Since the redefinition of SI base of operations units,^{[6] [vii]} the Coulomb abiding is no longer exactly divers and is discipline to the measurement error in the fine structure abiding, as calculated from CODATA 2018 recommended values being^[1]

${\displaystyle k_{\text{e}}=viii.987\,551\,7923\,(14)\times x^{9}\;\mathrm {kg{\cdot }grand^{three}{\cdot }south^{-4}{\cdot }A^{-ii}} .}$

Use [edit]

The Coulomb constant is used in many electric equations, although it is sometimes expressed as the post-obit product of the vacuum permittivity constant:

${\displaystyle k_{\text{e}}={\frac {1}{four\pi \varepsilon _{0}}}.}$

The Coulomb constant appears in many expressions including the following:

Coulomb's police

${\displaystyle \mathbf {F} =k_{\text{east}}{Qq \over r^{ii}}\mathbf {\hat {eastward}} _{r}.}$

Electric potential energy

${\displaystyle U_{\text{E}}(r)=k_{\text{eastward}}{\frac {Qq}{r}}.}$

Electric field

${\displaystyle \mathbf {Due east} =k_{\text{e}}\sum _{i=1}^{N}{\frac {Q_{i}}{r_{i}^{2}}}\mathbf {\hat {r}} _{i}.}$

See also [edit]

• Gravitational abiding
• Vacuum permittivity
• Vacuum permeability
• Inverse-square police force

References [edit]

1. ^ ^{a b} Derived from g _{due east} = ane/(4πε ₀) – "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Doubt. NIST. xx May 2019. Retrieved 2019-05-twenty .
2. ^ Britannica, The Editors of Encyclopaedia. "Coulomb". Encyclopedia Britannica . Retrieved 3 March 2021.
3. ^ Britannica, The Editors of Encyclopaedia. "Charles-Augustin de Coulomb". Encyclopedia Britannica . Retrieved 3 March 2021.
4. ^ Tomilin, K. (1999). "Fine-construction constant and dimension analysis". European Journal of Physics. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404.
5. ^ Coulomb's constant, HyperPhysics
6. ^ BIPM statement: Data for users nearly the proposed revision of the SI (PDF)
7. ^ "Decision CIPM/105-13 (Oct 2016)". The day is the 144th anniversary of the Metre Convention.

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Source: https://en.wikipedia.org/wiki/Coulomb_constant

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